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Thursday, April 11, 2013

Penrose Tiles

From Wikipedia,
Professor Penrose

Quasiperiodic tessellations are my favorite bit of recreational math. They came up in my thesis non-recreationally, and the best bit of Mathematica I ever wrote was to generate them for various different rotational symmetries.

A recent GeoGebra project got me using the RigidPolygon[] tool for the first time, and later I realized that this would enable me to make them in GeoGebra. The construction of the kites and darts is easy with the regular polygon tool. I couldn't make a tool to make copies, because it wanted too much information. I remembered the advice Kathryn Peake was giving to David Wees on Twitter for a sketch: make a button.

So the first sketch was the tiles without matching rules.

Purists, like Edmund Harriss (@gelada) in that Twitter conversation, will correctly point out that these are not Penrose tiles but they can be used to make a Penrose tiling.

Without the matching rules it's really easy to get yourself in trouble.

Here's the GeoGebraTube page. Unfortunately the applet doesn't work well in the HTML5 version, since you can make new tiles, but they can't be moved. Fine in the Java applet, though.

It took me a bit to figure out how to construct the tile alterations to provide the matching rules, but the upside is that it is customizable, so that now you can make the tiles into shapes that are pleasing to you. I can see how Dr. Penrose wound up with chickens, though.

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